Electrically tunable topological phase transition in non-Hermitian optical MEMS metasurfaces

Exceptional points (EPs), unique junctures in non-Hermitian open systems where eigenvalues and eigenstates simultaneously coalesce, have gained notable attention in photonics because of their enthralling physical principles and unique properties. Nonetheless, the experimental observation of EPs, particularly within the optical domain, has proven rather challenging because of the grueling demand for precise and comprehensive control over the parameter space, further compounded by the necessity for dynamic tunability. Here, we demonstrate the occurrence of optical EPs when operating with an electrically tunable non-Hermitian metasurface platform that synergizes chiral metasurfaces with piezoelectric MEMS mirrors. Moreover, we show that, with a carefully constructed metasurface, a voltage-controlled spectral space can be finely tuned to access not only the chiral EP but also the diabolic point characterized by degenerate eigenvalues and orthogonal eigenstates, thereby allowing for dynamic topological phase transition. Our work paves the way for developing cutting-edge optical devices rooted in EP physics and opening uncharted vistas in dynamic topological photonics.

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Supplementary Text Figs. S1 to S12 Legends for movies S1 and S2
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Supplementary Text
Design of the non-Hermitian optical MEMS metasurface To design the non-Hermitian metasurface possessing a chiral exceptional point (EP), we first consider a glass-gold-air-gold building block with only one gold nanobrick that is not rotated (e.g., the period is 250 nm).After sweeping the lateral dimensions of Lx and Ly, we select two elements that function as half-wave plates (HWPs) with equal reflection amplitudes but an intrinsic phase difference of π/2 at the design wavelength of λ = 810 nm with an air gap of ta = 430 nm (fig.S1,  A and B).The Jones matrices of these two HWPs in the circular polarization base are given by ) respectively, where rxx1 and rxx2 are the co-polarized reflection coefficients in the linear polarization base under x-polarized excitation.
The reflection matrix of HWP2 with a rotation angle of θ0 = 45° is expressed as By arranging these two HWPs in a unit cell with a period of p = 500 nm, the averaged Jones matrix can be approximately derived as shows the potential to design the chiral non-Hermitian metasurface.However, due to the near-field coupling between nanobricks within the designed unit cell, the metasurface does not show the expected performance at λ = 810 nm (fig.S1C).Therefore, the meta-atom dimensions, air gap, and wavelength should be iteratively optimized to realize chiral EP singularity.On the contrary, the diabolic point can still be observed near the wavelength of ⁓ 812 nm with the air gap of ta = 357 nm (fig.S1D), where the metasurface is located right at the nodes of the standing wave with zero illumination intensity.The outer electrodes enable quasi-linear movement for all voltages ranging from 0 V to 12 V, while the inner electrodes exhibit a nonlinear response for large voltages above 8 V.The estimated air gaps ta at the voltages of Vm = 1.9 V, 2.1 V, 2.6 V, and 2.7 V are ⁓ 2124.9 nm, ⁓ 2109.6 nm, ⁓ 2071.5 nm, and ⁓ 2063.9 nm, respectively.

Fig. S1 .
Fig. S1.Simulation of the as-designed non-Hermitian optical metasurfaces.(A and B) Simulated complex reflection coefficients of the glass-gold-air-gold building block with only one nanobrick (period is 250 nm) as a function of nanobrick dimensions of Lx and Ly in the linear polarization base.The other parameters are as follows: λ = 810 nm, tm = 50 nm, and ta = 430 nm.The nanobricks are rounded with radii of 15 nm (A) and 30 nm (B), respectively.|rxx| is the reflection amplitude under x-polarized excitation, and φxx and φyy represent the corresponding phase shifts under x-and y-polarized excitations.(C and D) Simulated coefficients of the reflection matrix as a function of wavelength at air gaps of ta = 430 nm (C) and 357 nm (D).The geometric dimensions of the unit cell (same configuration as that in Fig.1B) are set to l1 = 232 nm, w1 = 135 nm, l2 = 168 nm, w2 = 79 nm, θ = 45°, p = 500 nm, and tm = 50 nm.The corners of small and large nanobricks are rounded with radii of 30 nm and 15 nm, respectively.

Fig. S2 .
Fig. S2.Simulated amplitudes (A and C) and phase (B and D) of the reflection matrix eigenvalues in the geometrical parameter space Ω = [l1, w1] (A and B) and [l2, w2] (C and D).Self-intersecting Riemann surfaces are observed.The wavelength and air gap are set as 811.622 nm and 430.9 nm, respectively.

Fig. S3 .
Fig. S3.Simulated amplitudes (A) and phase (B) of the reflection matrix eigenvalues in the geometrical parameter space Ω = [ta, λ].A self-intersecting Riemann surface is observed.The dimensions of the chiral meta-atom are the same as those in the main text.

Fig. S5 .
Fig. S5.Estimated air gap as a function of the voltage applied to the outer (A) and inner (B)electrodes.The outer electrodes enable quasi-linear movement for all voltages ranging from 0 V to 12 V, while the inner electrodes exhibit a nonlinear response for large voltages above 8 V.

Fig. S6 .
Fig. S6.Measured polarization-resolved reflectance RLL (A), RLR (B), RRL (C), and RRR (D) under LCP and RCP incidence when four outer electrodes are actuated from 0 V to 12 V in a step of 0.1 V.

Fig. S9 .
Fig. S9.Experimental observation of the dynamic topological transition from a chiral EP to a DP at λ = 850.991nm when four outer electrodes are actuated.(A to C) Measured eigenvalues (A and B) and reflectance (C) as a function of wavelength at two different voltages of Vm = 10.1 V and 10.3 V when driving four outer electrodes.Anti-crossing of eigenvalue amplitudes and crossing of eigenvalue phases are observed for Vm = 10.1 V, while crossing of eigenvalue amplitudes and anti-crossing of eigenvalue phases are observed for Vm = 10.3V, revealing a chiral EP singularity at λ = 850.991nm for Vm between 10.1 V and 10.3 V. (D to F) Measured eigenvalues (D and E) and reflectance (F) as a function of wavelength at two different voltages of Vm = 9.4 V and 9.5 V. Crossing of eigenvalue amplitudes is observed for Vm = 9.4 V, indicating a DP at λ = 850.991nm for Vm of ⁓ 9.4 V.The estimated air gaps ta at the voltages of Vm = 9.4 V, 9.5 V, 10.1 V, and 10.3 V are ⁓ 2913.4 nm, ⁓ 2920.3 nm, ⁓ 2966.0 nm, and ⁓ 2981.1 nm, respectively.

Fig. S10 .
Fig. S10.Experimental observation of the dynamic topological transition from a chiral EP to a DP at λ = 849.505nm when four inner electrodes are actuated.(A to C) Measured eigenvalues (A and B) and reflectance (C) as a function of wavelength at two different voltages of Vm = 1.9 V and 2.1 V. Crossing of eigenvalue amplitudes and anti-crossing of eigenvalue phases are observed for Vm = 1.9 V, while anti-crossing of eigenvalue amplitudes and crossing of eigenvalue phases are observed for Vm = 2.1 V, revealing a chiral EP singularity at λ = 849.505nm for Vm between 1.9 V and 2.1 V. (D to F) Measured eigenvalues (D and E) and reflectance (F) as a function of wavelength at two different voltages of Vm = 2.6 V and 2.7 V. Crossing of eigenvalue amplitudes is observed for Vm = 2.7 V, indicating a DP at λ = 849.505nm for Vm of ⁓ 2.7 V. The estimated air gaps ta at the voltages of Vm = 1.9 V, 2.1 V, 2.6 V, and 2.7 V are ⁓ 2124.9 nm, ⁓ 2109.6 nm, ⁓ 2071.5 nm, and ⁓ 2063.9 nm, respectively.

Fig. S11 .
Fig. S11.Voltage-controlled polarization evolution at λ = 849.505nm when four inner electrodes are actuated.(A) Measured reflectance as a function of the applied voltage.(B) Voltage-controlled polarization trajectory mapped on the Poincaré sphere for RCP incidence.

Fig. S12 .
Fig.S12.Sensing with the non-Hermitian MEMS metasurface.The simulated absolute value of the difference of two eigenvalues as a function of the refractive index of the chemical that fills the gap between the chiral gold array and MEMS mirror.The EP mode shows higher sensitivity for ultra-small refractive index changes, while the DP mode is more sensitive to large changes.We could selectively drive the non-Hermitian MEMS sensor to work in the EP or DP mode to detect chemicals at any concentration with high sensitivity.The wavelength, air gap, and dimensions of the chiral meta-atom are the same as those in the main text.